Role of the Λ(1600) in the K − p→Λ π 0 π 0 reaction
aa r X i v : . [ h e p - ph ] A p r Role of the
Λ(1600) in the K − p → Λ π π reaction He Zhou
1, 2 and Ju-Jun Xie
1, 2, 3, ∗ Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China School of Nuclear Sciences and Technology, University of Chinese Academy of Sciences, Beijing 101408, China School of Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China (Dated: April 9, 2020)Role of the Λ(1600) is studied in the K − p → Λ π π reaction by using the effective Lagrangianapproach near the threshold. We perform a calculation for the total and differential cross sections byconsidering the contributions from the Λ(1600) and Λ(1670) intermediate resonances decaying into π Σ ∗ (1385) with Σ ∗ (1385) decaying into π Λ. Besides, the non-resonance process from u -channelnucleon pole is also taken into account. With our model parameters, the current experimentaldata on the total cross sections of the K − p → Λ π π reaction can be well reproduced. It isshown that we really need the contribution from the Λ(1600) with spin-parity J P = 1 / + , andthat these measurements can be used to determine some of the properties of the Λ(1600) resonance.Furthermore, we also plot the π Λ invariant mass distributions which could be tested by the futureexperimental measurements.
Key words : ¯ KN scattering; Effective Lagrangian approach; Hyperon resonance. I. INTRODUCTION
The ¯ KN scattering has been widely used to studythe properties of the hyperon resonances [1–9], and itis extremely important to investigate these low excitedhyperon states through the proposed K L beam experi-ments at Jefferson Lab [10, 11]. By using a chiral uni-tary approach [12–15], the meson-baryon interactions areinvestigated and it was found that there are two polesin the neighbourhood of the well established Λ(1405)state, which is actually a superposition of these two J P = 1 / − resonances. Recently, within a dynami-cal coupled-channels model [16, 17], some hyperon reso-nance parameters are extracted through a comprehensivepartial-wave analysis of the K − p → ¯ KN , π Σ, π Λ, η Λ,and K Ξ data up to invariant mass W = 2 . J P = 3 / + is also predicted in Refs. [16, 17]. On thecontrary, Liu and Xie [18–20] analyzed the K − p → η Λreaction [21] with an effective Lagrangian approach andimplied a new Λ resonance with J P = 3 / − . Its mass isabout 1670 MeV but its width is much small comparedwith the one of the well established Λ(1690) resonance.Thus there are still some ambiguities of the Λ excitedstates needs to be clarified.On the experimental side, the Crystal Ball Collabo-ration reported the measurements with high precisionof the K − p → Λ π π reaction at eight incidents of K − momenta between 514 and 750 MeV, correspond-ing to center of mass (c.m.) energies from 1569 to 1676MeV [22]. It is shown that this reaction is dominatedby the π Σ ∗ (1385) intermediate state in s -channel, andthe contribution of the f (500) meson in t -channel to the K − p → Λ π π reaction appears to be very small and canbe neglected. Indeed, it is shown that the contribution ∗ Electronic address: [email protected] of scalar meson f (500) and f (980) from the K + K − → π π transition term is negligible [23, 24]. In addition,the strength of the total cross section of K − p → Λ π π reaction could be well reproduced in terms of the largecoupling of Λ(1520) to π Σ ∗ (1385), which is a prediction ofthe chiral unitary approach [24, 25]. On the other hand,with the aim for searching for the evidence for the possi-ble Σ excited state with J P = 1 / − , which was predictedwithin the unquenched penta-quark models [26, 27], the K − p → Λ π + π − reaction was investigated at the energyregion of the Λ(1520) resonance peak by using the ef-fective Lagrangian approach [28], where it is found thatthere is evidence for the existence of the new Σ ∗ state inthe K − p → Λ π + π − reaction.For the K − p → Λ π π reaction, the main contri-bution is from the Λ ∗ resonance through the process K − p → Λ ∗ → π Σ ∗ (1385) → π π Λ. This reactiongives us a rather clean platform to study the isospin-0Λ ∗ resonances because there are no isospin-1 Σ ∗ reso-nances that contribute to K − p → π Σ ∗ (1385). In theenergy region of the current experimental measurementsby the Crystal Ball Collaboration [22], there are two wellestablished Λ ∗ resonances give significant contributions:the three-star Λ(1600) with J P = 1 / + and the four-starΛ(1670) with J P = 1 / − . Their Breit-Wigner massesand widths are [29]: M Λ ∗ = 1560 ∼ , Γ Λ ∗ = 50 ∼ , (1) M Λ ∗ = 1660 ∼ , Γ Λ ∗ = 25 ∼ , (2)all in units of MeV and for which we hand used the no-tation Λ ∗ and Λ ∗ to refer to the Λ(1600) and Λ(1670)resonances, respectively. It is interesting to notice thatboth the mass and width of the Λ(1600) resonance arewith large uncertainties, while the ones for Λ(1670) res-onance are much precise. Furthermore, in the work ofRef. [7], the most precise data on the K − p → π Σ reac-tion were analyzed to study the Λ ∗ resonances, and it isfound that the Λ(1600) resonance is definitely needed.The fitted resonance parameters for the Λ(1600) are M Λ(1600) = 1574 . ± . Λ(1600) = 81 . ± . K − p → Λ π π reaction. In fact, the energy dependence of the total crosssection of K − p → Λ π π reaction [22] has a broad shoul-der around the energy region of the Λ(1600) state.In the present work, based on the experimental mea-surements of the Crystal Ball Collaboration [22], westudy the role of the Λ(1600) and Λ(1670) resonancesin the K − p → Λ π π reaction within the effective La-grangian method and the resonance model. In addition,the non-resonance process from the u -channel nucleonpole is also considered as the background. Since thereare large uncertainties for the mass and width of theΛ(1600) resonance, we will vary them to reproduce theexperimental data. While for the Λ(1670) resonance, wetake the average values for its mass and width as quotedin the Particle Data Group (PDG) [29]. The total anddifferential cross sections of the K − p → Λ π π reactionare calculated. It is found that the contribution of theΛ(1600) resonance is significant, and the experimentaldata on the total cross sections and angular distributions,around the reaction energy region of the Λ(1600) state,can be well reproduced with the model parameters.The present paper is organized as follows: In sec. II,we discuss the formalism and the main ingredients forour theoretical calculations; In sec. III we present ournumerical results and conclusions; A short summary isgiven in the last section. II. FORMALISM AND INGREDIENTS
The combination of the resonance model and the ef-fective Lagrangian approach is an important theoreticaltool in describing the various scattering processes in theresonance production region [30–33]. In this section, weintroduce the theoretical formalism and ingredients tostudy the K − p → Λ π π reaction by using the effectiveLagrangian approach and resonance model. A. Feynman diagrams and effective interactionLagrangian densities
The basic tree-level Feynman diagrams for the K − p → Λ π π reaction are shown in Fig. 1. These include s -channel Λ ∗ resonances process [Fig. 1 (a)] and u -channelnucleon pole diagram [Fig. 1 (b)]. For the π Λ produc-tion, we consider only the contribution from Σ ∗ (1385).The t -channel K + exchange term via K + K − → π π transition is not considered since its contribution is rathersmall. Besides, the t -channel K ∗ exchange is also ne-glected since this mechanism is much suppressed due tothe highly off-shell effect of the K ∗ propagator when the π Λ invariant mass is close to the Σ ∗ (1385) mass. To evaluate the contributions of those terms shownin Fig. 1, the effective Lagrangian densities for relevantinteraction vertexes are needed. Following Refs. [34–40],the Lagrangian densities used in this work are, L Λ ∗ ¯ KN = − g Λ ∗ ¯ KN m N + M Λ ∗ ¯Λ ∗ γ γ µ ∂ µ φ ¯ K N + h . c ., (3) L Λ ∗ π Σ ∗ = g Λ ∗ π Σ ∗ m π ¯Σ ∗ µ ∂ µ ( ~τ · ~π )Λ ∗ + h . c ., (4) L Λ ∗ ¯ KN = g Λ ∗ ¯ KN ¯Λ ∗ φ ¯ K N + h . c ., (5) L Λ ∗ π Σ ∗ = g Λ ∗ π Σ ∗ m π ¯Σ ∗ µ γ ∂ µ ( ~τ · ~π )Λ ∗ + h . c ., (6) L π ΛΣ ∗ = g π ΛΣ ∗ m π ¯Σ ∗ µ ∂ µ ( ~τ · ~π )Λ + h . c ., (7) L ¯ KN Σ ∗ = g ¯ KN Σ ∗ m ¯ K ¯Σ ∗ µ ∂ µ φ ¯ K N + h . c ., (8) L πNN = − g πNN m N ¯ N γ γ µ ∂ µ ( ~τ · ~π ) N, (9)where Σ ∗ µ is the Rarita-Schwinger field of the Σ ∗ (1385)resonance with spin , and ~τ is a usual isospin-1/2 Paulimatrix operator.For the coupling constants in the above Lagrangiandensities for u -channel process, we take g πNN = 13 . g N ¯ K Σ ∗ = − .
19 which are used in previousworks [41–43] for studying different processes. For thecoupling constant g π ΛΣ ∗ (1385) and g Λ(1670) ¯ KN , they canbe determined from the experimental observed partialdecay widths of Σ ∗ (1385) → π Λ and Λ(1670) → ¯ KN ,respectively.With the effective interaction Lagrangians describedby Eqs. (3), (5), and (7), the partial decay widthsΓ Σ ∗ → π Λ and Γ Λ ∗ → ¯ KN can be easily obtained [29]. Thecoupling constants related to the partial decay widths arewritten as,Γ Λ ∗ → ¯ KN = g ∗ ¯ KN π ( E N − m N ) p ¯ KN M Λ ∗ , (10)Γ Λ ∗ → ¯ KN = g ∗ ¯ KN π ( E N + m N ) p ¯ KN M Λ ∗ , (11)Γ Σ ∗ → π Λ = g ∗ π Λ π ( E Λ + m Λ ) p π Λ m π m Σ ∗ , (12)with E N = M ∗ / Λ ∗ + m N − m K M Λ ∗ / Λ ∗ , (13) p ¯ KN = q E N − m N , (14) E Λ = m ∗ + m − m π m Σ ∗ , (15) p π Λ = q E − m . (16)With the masses, widths and branching ratios of Λ(1670)and Σ ∗ (1385) resonances quoting in PDG [29], the nu-merical results for the relevant coupling constants are K − p Λ ∗ π π ΛΣ ∗ p K − π p Σ ∗ π Λ p p q s q Σ ∗ p p p p p q u q Σ ∗ p p p ( a ) ( b ) FIG. 1: Feynman diagrams of the K − p → π π Λ reaction. The contributions from s -channel Λ(1600) and Λ(1670) resonancesand u -channel nucleon pole are considered. We also show the definition of the kinematical ( p , p , p , p , p ) variables that weuse in the present calculation. In addition, we use q s = p + p , q u = p − p , and q Σ ∗ = p + p . listed in Table I, while the other coupling constantsneeded in this work will be discussed below. TABLE I: Relevant parameters used in the present calcula-tion. The masses, widths and branching ratios of Λ(1670)and Σ ∗ (1385) resonances are taken from PDG [29], while forthe Λ(1600) resonance, these values are determined to theexperimental data.State ( J P ) Mass Width Decay Branching g / π (MeV) (MeV) mode ratio (%)Σ ∗ (1385) (
32 + ) 1385 37 π Λ 87 0.12Λ(1670) ( − ) 1670 35 ¯ KN
25 0.009 π Σ ∗ (1385) 22.4 1.07Λ(1600) (
12 + ) 1580 150 ¯ KN π Σ ∗ (1385) 6.5 0.05 B. Propagators and form factors
To get the scattering amplitude of the K − p → Λ π π reaction corresponding to the Feynman diagrams shownin Fig. 1, we also need the propagators for spin parti-cles: nucleon, Λ(1600) and Λ(1670), and Σ ∗ (1385) reso-nance with spin , G p ( q u ) = i / q u + m N q u − m p , (17) G Λ ∗ / Λ ∗ ( q s ) = i / q s + M Λ ∗ / Λ ∗ q s − M ∗ / Λ ∗ + iM Λ ∗ / Λ ∗ Γ Λ ∗ / Λ ∗ , (18) G µν Σ ∗ ( q Σ ∗ ) = i (/ q Σ ∗ + m Σ ∗ ) P µν ( q Σ ∗ ) q ∗ − m ∗ + im Σ ∗ Γ Σ ∗ , (19)with P µν ( q Σ ∗ ) = − g µν + 13 γ µ γ ν + 23 q µ Σ ∗ q ν Σ ∗ m ∗ + 13 m Σ ∗ ( γ µ q ν Σ ∗ − γ ν q µ Σ ∗ ) , (20) where q u , q s and q Σ ∗ are the momenta of nucleon polein u -channel, Λ(1600) or Λ(1670) resonance in s -channel,and Σ ∗ (1385) resonance, respectively.Finally, we need to also include the off-shell form fac-tors in the scattering amplitudes. There is no uniquetheoretical way to introduce the form factors, hence, weadopt here the common scheme used in many previousworks [42–44], f i = Λ i Λ i + ( q i − M i ) , i = s, u, Σ ∗ (21)with q s = s, q u = u, q ∗ = M π Λ M u = m N , M Σ ∗ = m Σ ∗ ,M s = M Λ ∗ / Λ ∗ , , (22)where s and u are the Lorentz-invariant Mandelstam vari-ables, while M π Λ is the invariant mass of the π Λ sys-tem. In the present calculation, q s = p + p , q u = p − p ,and q Σ ∗ = p + p are the 4-momenta of intermediateΛ(1600) or Λ(1670) resonance, exchanged nucleon polein the u -channel, and the Σ ∗ (1385) resonance decayinginto π Λ, respectively, while p , p , p , p , and p are the4-momenta for K − , p , π , π , and Λ, respectively. Be-sides, we will consider same cut-off values for the back-ground and resonant terms, i.e. Λ s = Λ u . Note that thenumerical results are not sensitive to Λ s and Λ Σ ∗ . C. Scattering amplitudes
With the effective interaction Lagrangian densitiesgiven above, we can easily construct the invariant scat-tering amplitudes for the K − p → Λ π π reaction corre-sponding to the diagrams shown in Fig. 1: M = M (Λ ∗ ) + M (Λ ∗ ) + M ( N ) . (23)Each of the above amplitudes can be obtained straight-forwardly as, M ( i ) = ¯ u ( p , s Λ ) G µν Σ ∗ A µν ( i ) u ( p , s p ) , (24)where s Λ and s p are the spin polarization variables forthe final Λ and initial proton, respectively. The reduced A µν ( i ) can be also easily obtained: A µν (Λ ∗ ) = − ig p µ p ν G Λ ∗ ( q s ) γ / p f s (Λ ∗ ) f Σ ∗ , (25) A µν (Λ ∗ ) = g p µ p ν γ G Λ ∗ ( q s ) f s (Λ ∗ ) f Σ ∗ , (26) A µν ( N ) = − ig p µ p ν G N ( q u ) γ / p f u f Σ ∗ , (27)with g = g Σ ∗ π Λ g Λ ∗ π Σ ∗ g Λ ∗ ¯ KN m π ( m N + M Λ ∗ ) , (28) g = g Σ ∗ π Λ g Λ ∗ π Σ ∗ g Λ ∗ ¯ KN m π , (29) g = g Σ ∗ π Λ g Σ ∗ ¯ KN g πNN m π m ¯ K m N . (30)Then, the cross section for the K − p → Λ π π reactioncan be calculated by [29, 45] dσ = 14 1(2 π ) m p q ( p · p ) − m p m K − × X s p ,s Λ |M| × d p E d p E m Λ d p E δ ( p + p − p − p − p )= 12 π m p m Λ q s [( p · p ) − m p m K − ] X s p ,s Λ |M| (31) ×| ~p || ~p ∗ | dM π Λ d Ω d Ω ∗ , (32)with s = ( p + p ) = m p + m K − + 2 p · p , and ~p ∗ andΩ ∗ are the three-momentum and solid angle of the outgoing Λ in the center-of-mass (c.m.) frame of the final π Λ system, while ~p and Ω are the three-momentumand solid angle of the π meson in the c.m. frame ofthe initial K − p system. Note that we have already takeninto account the factor 1 / III. NUMERICAL RESULTS ANDDISCUSSIONS
The theoretical results for the total cross sections forbeam momenta p K − (module of the three momentum ~p ) from 0 . . u -channel process in describing the total cross sections.The contributions from different mechanisms are shownseparately. The red dashed, blue dotted, and green dash-dotted curves stand for contributions from the Λ(1600),Λ(1670) and u -channel, respectively. Their total con-tributions are shown by the solid line. The theoretical Note that the total squared amplitude for K − p → π π Λ reac-tion is symmetrized in the momenta p and p to account for thetwo π in the final state. numerical results are obtained with the following param-eters: Λ s = 600 MeV for the Λ(1600) and Λ(1670) res-onances, Λ u = Λ Σ ∗ = 600 MeV, M Λ ∗ = 1580 MeV,Γ Λ ∗ = 150 MeV, g Λ ∗ π Σ ∗ = 0 .
79, and g Λ ∗ π Σ ∗ = 3 . s ( m b) P K - (GeV) Total (1600) (1670) u channel FIG. 2: Theoretical results of the total cross sections of K − p → Λ π π reaction. The experimental data are takenfrom Ref. [22]. From Fig. 2, one can see that we can fairly well re-produce the experimental data of Ref. [22], and that theΛ(1600) resonance gives a dominant contribution to thereaction around p K − = 630 MeV, while the contributionof Λ(1670) is significant around p K − = 750 MeV. On theother hand, it is seen clearly that the inclusion of theΛ(1600) resonance is crucial to achieve a fairly good de-scription of the experimental data. However, we can notdescribe the enhancement at low energy region, where itcould be explained by the tail of the contribution of theΛ(1520) in Refs. [24, 25], and it may also be explainedby the possible Σ ∗ (1380) → π Λ in s wave as proposedin Ref. [28]. Such calculations are beyond the scope ofthe present investigation but we will clarify this issue ina future study.With the obtained strong coupling constants g Λ ∗ π Σ ∗ and g Λ ∗ π Σ ∗ , we have evaluated the Λ(1600) and Λ(1670)resonances to the π Σ ∗ (1385) partial decay width:Γ Λ ∗ / Λ ∗ → π Σ ∗ = g ∗ / Λ ∗ π Σ ∗ M Λ ∗ / Λ ∗ πm π m Σ ∗ ( E Σ ∗ ± m Σ ∗ ) p π Σ ∗ , with E Σ ∗ = M ∗ / Λ ∗ + m ∗ − m π M Λ ∗ / Λ ∗ , (33) p π Σ ∗ = q E ∗ − m ∗ , (34)as deduced from the Lagrangians of Eq. (4) and Eq. (6).With the partial decay widths, we can then obtain thebranching ratios. The numerical predictions for these -1.0 -0.5 0.0 0.5 1.00.000.020.040.060.080.10 d s /d W (mb/sr) cos q ( p ) Total (1600) (1670) u channel -1.0 -0.5 0.0 0.5 1.00.000.020.040.060.080.100.12 d s /d W (mb/sr) cos q ( p ) Total (1600) (1670) u channel -1.0 -0.5 0.0 0.5 1.00.000.020.040.060.080.10 d s /d W (mb/sr) cos q ( p ) Total (1600) (1670) u channel FIG. 3: Angular differential cross sections for the K − p → Λ π π reaction as a function of cos θ with θ the angel be-tween the π direction and the beam direction in the overallc.m. system at p K − = 581 (up), 629 (middle), and 687 MeV(down). The experimental data are taken from Ref. [22]. branching ratios are also given in Table I. Note that theuncertainties of the coupling constants and cut off param-eters are not studied in this work, since, including sucheffects, the scattering amplitudes would be more complexdue to additional model parameters, and we cannot ex-actly determine these parameters. Thus, we leave theseinvestigations to further studies when more precise ex-perimental measurements become available.In addition to the total cross sections, we also compute the angle distributions for K − p → Λ π π reaction. Thecorresponding theoretically numerical results at p K − =581, 629, and 687 MeV, where the contribution of theΛ(1600) resonance is dominant, are shown in Fig. 3. Forcomparison, we also show the experimental data fromRef. [22]. It is obvious that we can fairly well reproducethe current experimental data on the angular distributionof the K − p → Λ π π reaction thanks to the contributionof the Λ(1600) resonance. d s /d M p L ( m b/GeV) M p L (GeV) Total (1600) (1670) u channel d s /d M p L ( m b/GeV) M p L (GeV) Total (1600) (1670) u channel d s /d M p L ( m b/GeV) M p L (GeV) Total (1600) (1670) u channel FIG. 4: The π Λ invariant mass distribution of K − p → Λ π π reaction at p K − = 581 (up), 629 (middle), and 687MeV (down). Finally, in Fig. 4, we show the theoretical results onthe differential cross section dσ/dM π Λ as a function ofthe invariant mass of a pair of π Λ for the values of K − momentum, 581, 629 and 687 MeV. From these figures,we see that the shape of the π Λ invariant mass distri-butions are different with the beam energy increasing.We hope that the future experimental measurements cancheck our model calculations.
IV. SUMMARY
In summary, we have investigated the total and dif-ferential cross sections of the K − p → Λ π π reactionwithin an effective Lagrangian approach and the res-onance model. The role played by the Λ(1600) andΛ(1670) resonances are studied. It is shown that ourmodel calculations lead to a fair description of the exper-imental data on the total cross section except for the lowenergy date. The scheme proposed herein should be sup-plemented with some other reaction mechanisms whichcould improve the achieved description of the low energyenhancement. Indeed, as is proposed in Refs. [24, 25] theΛ(1520) plays an important role in the K − p → Λ π π reaction with the K − p → π Σ ∗ (1385) amplitude obtainedfrom the chiral unitary approach. However, we haveshown here that the Λ(1600) and Λ(1670) resonancesgive dominant contributions, and the consideration of theΛ(1600) resonance is crucial.Finally, we would like to stress that, thanks to theimportant role played by the resonant contribution ofΛ(1600) resonance in the K − p → Λ π π reaction, wecan describe experimental data on the total cross sectionand angle distributions. Accurate data for this reactioncan be used to improve our knowledge of some Λ(1600)properties, which are at present poorly known. This workconstitutes a first step in this direction. Acknowledgements
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